Class 6 Mathematics Notes Chapter 6 (Integers) – Mathematics Book

Detailed Notes with MCQs of Chapter 6: Integers from your NCERT Class 6 Mathematics book. This is a foundational chapter, and understanding it well is crucial not just for your school exams but also forms the base for many concepts tested in government exams. Pay close attention!

Chapter 6: Integers - Detailed Notes for Exam Preparation

1. Introduction: Why Do We Need Integers?

• We already know about Natural Numbers (counting numbers: 1, 2, 3, ...) and Whole Numbers (Natural numbers including zero: 0, 1, 2, 3, ...).
• However, whole numbers are not sufficient to represent situations like:
  • Temperatures below 0°C (like -5°C).
  • Depths below sea level (like -100 meters).
  • Financial loss or debt (like a loss of ₹500, represented as -500).
  • Scores in games where points can be deducted.
• To represent these 'opposite' quantities, we need Negative Numbers.

2. What are Integers?

• Integers are the collection of all whole numbers and the negative of natural numbers.
• The set of integers is represented by the symbol Z (from the German word 'Zahlen' meaning numbers).
• Integers include: {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}
• Positive Integers: 1, 2, 3, 4, ... (These are the same as Natural Numbers).
• Negative Integers: -1, -2, -3, -4, ...
• Zero (0): Zero is an integer, but it is neither positive nor negative.

3. Representing Integers on a Number Line

• A number line is a visual way to represent integers.

• Draw a straight line. Mark a point in the middle and label it 0 (Zero).

• Mark points at equal distances to the right of 0. Label them 1, 2, 3, ... (Positive Integers).

• Mark points at equal distances to the left of 0. Label them -1, -2, -3, ... (Negative Integers).

  <--|---|---|---|---|---|---|---|---|---|---|-->
  ... -5 -4 -3 -2 -1 0 1 2 3 4 5 ...

• Key Points about the Number Line:

  • Numbers increase as we move to the right.
  • Numbers decrease as we move to the left.
  • Every integer has a unique position on the number line.

4. Ordering of Integers (Comparison)

• Using the number line helps compare integers:
  • Any integer to the right of another integer is greater.
  • Any integer to the left of another integer is smaller.
• Rules for Comparison:
  • Every positive integer is greater than 0. (e.g., 5 > 0)
  • Every negative integer is less than 0. (e.g., -3 < 0)
  • Every positive integer is greater than every negative integer. (e.g., 2 > -10)
  • When comparing two positive integers, the one with the larger numerical value is greater. (e.g., 8 > 3)
  • When comparing two negative integers, the one closer to zero (further to the right on the number line) is greater. This is a common point of confusion! (e.g., -2 > -5, because -2 is to the right of -5).

5. Operations on Integers

(a) Addition of Integers

• Using Number Line:

  • To add a positive integer, move to the right on the number line.
    • Example: 2 + 3 = ? Start at 2, move 3 steps right -> reach 5. So, 2 + 3 = 5.
    • Example: (-1) + 4 = ? Start at -1, move 4 steps right -> reach 3. So, (-1) + 4 = 3.
  • To add a negative integer, move to the left on the number line.
    • Example: 4 + (-2) = ? Start at 4, move 2 steps left -> reach 2. So, 4 + (-2) = 2.
    • Example: (-2) + (-3) = ? Start at -2, move 3 steps left -> reach -5. So, (-2) + (-3) = -5.

• Rules for Addition (Without Number Line):

  • Same Signs: Add the numerical values (ignoring the sign) and keep the common sign.
    • (+ve) + (+ve) = +ve (e.g., 5 + 3 = 8)
    • (-ve) + (-ve) = -ve (e.g., (-5) + (-3) = -8)
  • Different Signs: Subtract the smaller numerical value from the larger numerical value (ignoring the signs). Give the result the sign of the integer with the larger numerical value.
    • e.g., (-7) + 4 = ? (Numerical values are 7 and 4. 7 > 4. Difference is 7-4=3. Sign of larger value (-7) is negative). So, (-7) + 4 = -3.
    • e.g., 9 + (-5) = ? (Numerical values are 9 and 5. 9 > 5. Difference is 9-5=4. Sign of larger value (9) is positive). So, 9 + (-5) = 4.

(b) Subtraction of Integers

• Key Concept: Subtracting an integer is the same as adding its additive inverse.

• Additive Inverse: The additive inverse of an integer is the integer that, when added to it, gives zero.

  • Additive inverse of 5 is -5 (because 5 + (-5) = 0).
  • Additive inverse of -8 is 8 (because (-8) + 8 = 0).
  • Additive inverse of 'a' is '-a'. Additive inverse of '-a' is 'a'.

• Rule for Subtraction: To subtract an integer 'b' from an integer 'a', change the sign of 'b' and add it to 'a'.

  • a - b = a + (additive inverse of b) = a + (-b)
  • a - (-b) = a + (additive inverse of -b) = a + b

• Examples:

  • 8 - 3 = 8 + (-3) = 5 (Using addition rules for different signs)
  • 6 - (-4) = 6 + 4 = 10
  • (-5) - 2 = (-5) + (-2) = -7 (Using addition rules for same signs)
  • (-7) - (-3) = (-7) + 3 = -4 (Using addition rules for different signs)

6. Important Points to Remember for Exams

• 0 is an integer, neither positive nor negative.
• The smallest positive integer is 1. There is no largest positive integer.
• The greatest negative integer is -1. There is no smallest negative integer.
• Be careful when comparing negative numbers (-2 is greater than -10).
• Subtraction is equivalent to adding the additive inverse. Master the sign rules for addition.
• Understand the application of integers in real-world contexts (temperature, elevation, profit/loss).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts we just discussed. Try to solve them yourself!

1. Which of the following is the greatest negative integer?
   (a) -100
   (b) -1
   (c) 0
   (d) -10

2. The integer representing a depth of 500 meters below sea level is:
   (a) 500
   (b) -500
   (c) 0
   (d) +500m

3. On the number line, the integer -3 lies:
   (a) To the right of 0
   (b) To the left of -4
   (c) To the right of -2
   (d) To the left of -2

4. What is the additive inverse of -7?
   (a) -7
   (b) 7
   (c) 0
   (d) 1/7

5. Calculate the value of: (-15) + 8
   (a) -23
   (b) 23
   (c) -7
   (d) 7

6. Calculate the value of: 9 - (-4)
   (a) 5
   (b) -5
   (c) 13
   (d) -13

7. Which comparison is correct?
   (a) -8 > -2
   (b) -2 < -8
   (c) -8 < -2
   (d) -8 = -2

8. What must be subtracted from -3 to get -9?
   (a) -6
   (b) 6
   (c) -12
   (d) 12

9. The sum of two integers is -10. If one of them is 5, the other is:
   (a) 5
   (b) -5
   (c) 15
   (d) -15

10. Which integer is 4 less than -1?
    (a) 3
    (b) -3
    (c) 5
    (d) -5

Answers to MCQs:

1. (b) -1
2. (b) -500
3. (d) To the left of -2
4. (b) 7
5. (c) -7
6. (c) 13
7. (c) -8 < -2
8. (b) 6 [Let the number be x. (-3) - x = -9 => -3 + 9 = x => x = 6]
9. (d) -15 [Let the other integer be y. 5 + y = -10 => y = -10 - 5 => y = -15]
10. (d) -5 [-1 minus 4 means (-1) - 4 = (-1) + (-4) = -5]

Make sure you understand the reasoning behind each answer. Practice more problems from your textbook and other resources. Good luck with your preparation!